You have not told us where the students are coming from but it is pretty safe to say that these days no high school students (with exception of few countries on the world) are exposed to any meaningful Geometry course.
How about teaching them some real old fashion synthetic (Euclidean/Lobachevsky) geometry course based let say on Kiselev's classic
with possible excursion into Projective geometry.
There is no more natural place to introduce the concept of groups (actions on the sets) than in Geometry (composition of isometric transformations). There is no more natural place to introduce them to the concept of measure. It is very easy to involve hard combinatorial problems and many other things. Finally, set theory is all over the geometry and axiomatic method rules.
Many of the most challenging "Olympic problems" are geometric in its nature.